We know that $ 0<\dfrac{n^2-2}{n^3+n} <\dfrac{n^2-2}{n^3} < \dfrac{n^2}{n^3} = \dfrac{1}{n}$ for any $n\ge 2$. Considering this fact, what does the direct comparison test say about $\sum_{n=2}^{\infty }\dfrac{n^2-2}{n^3+n}$ ? Choose 1 answer: Choose 1 answer: (Choice A) A The series converges. (Choice B) B The series diverges. (Choice C) C The test is inconclusive.
Explanation: $\sum\limits_{n=2}^{\infty }~{\frac{1}{{n}}}~$ is the harmonic series which is known to diverge. Because our given series is term-by-term less than a divergent series, the direct comparison test does not apply. So the direct comparison test is inconclusive.